divergent double sequences and series* · of the sequence xm,n by the transformation. it is a well...

DIVERGENT DOUBLE SEQUENCES AND SERIES*

BY

G. M. ROBISON

I. Introduction

Several definitions for giving a value to a divergent simple series, as

for example the Cesàro's and Holder's means, can be expressed by means

of a linear transformation defined by an infinite matrix on numbers. Two

types of these transformations are given as follows, one by a triangular

matrix, the other by a square matrix.

5 :

fll.t ßl.2 #1.3 fll.4 fll.S

Û2.I #2,2 #2,3 Ö2.4 #2,5

03.1 Ö3.2 03,3 Ö3.4 Ö3.6

For any given sequence {xn} a new sequence {yn} is defined as follows :

n

Vn= 2 a».* **, f°r the matrix T ,k-l

yn= 2^i an.h Xk, for the matrix 5 ,*=1

provided in the latter case y„ has a meaning. If to any matrix of type T

we adjoin the elements a„,* = 0, k>n (all »), we obtain a matrix of type 5.

Since this addition does not affect the transformation, any transformation

of the type T may be considered as a special case of a transformation of

type 5. If for either transformation limn^n exists, the limit is called

the generalized value of the sequence xn by the transformation. If when-

ever xn converges, y„ converges to the same value, then the transformation

* Presented to the Society, December 29, 1920; received by the editors in March, 1922.

50

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DIVERGENT DOUBLE SEQUENCES 51

is said to be regular. The criterion for regularity of these transformations

is stated as follows :

Theorem.* A necessary and sufficient condition that the transformation T

be regular is that

(a) lim a„,k = 0for every k,B—.00

n

(b) lim E an,k—l,

n

(c) E |aB.*|<;4/0r all n ,

*=i(a) Iim an,k = Ofor each k,

B-»oo

Theorem, f A necessary and sufficient condition that the transformation S

be regular is that

CO

(b) E Ia»,* I converge for each n,k-i

CO

W E \an,k\<Aforalln,fc-i

03

(d) lim E a„.k=l .n-»» i-i

Corresponding to these definitions of summability for a single series,

we have the following definitions for giving a value to a divergent double

series. Let the given series be represented as follows :

Wl,l + «1,2 + «1,3 + «1,4 + «1,6+ • • •

+ M2,l + «2.2 + «2,3 + «2,4 + «2,5+ ' • '

+ M3,l + M3,2 + «3,3 + M3,4+ ' ' •

+«.;

then the double sequence xm,n for this series is given by the following

equality :m,B

Xm,n~ E MM •A—1,1-1

* L. L. Silvermann, Missouri dissertation, 1910; ToepUtz, PraceMatematyczno-

Fizyczne, vol.22 (1911), p. 113.f This theorem was given in the classroom by Professor Hurwitz at Cornell University 1917-18.

Published statements of proofs are due to T. H. Hildebrandt, Bulletin of the American Ma-

thematical Society, vol. 24 (1917-18), p. 429; R. D. Carmichael, Bulletin of the Ameri-

can Mathematical Society, vol. 25 (1918-19), p. 118; I. Schur, Journal für die reine

und angewandte Mathematik, vol. 151 (1920), p. 79.


52 G. M. ROBISON [January

Thus we have also

Wm1n=:^m,»T^n-l1n-l Xm,n—1 Xm,x—n, m>l, «>I J

Ux,n=Xx,n-Xi.n-ln,n>l )

«m,i = »m,i—*»-i.i, m>l ;

Ui,i = Xi,i.

We define a new sequence by the relation

*n,n

y»i,n= ¿j am,n,k,i Xk.ll ,

k—1,1-1

We shall call this transformation and its matrix A : (am¡n,k,i) of the

type T ; here k = m; l = n. Again we may write

00 rCO

ym,n= 2-1 am,n,k,l Xk,ll .k-l.l-1

provided ym,n has a meaning. We shall call this transformation and its

matrix A : (am.n,k,i) of the type S ; here k and I take on all positive integral

values. Any transformation of type T may be considered as a special case

of a transformation of type 5 ; for by adding the elements

am,n,k,i = 0,m<k,n<l, all wand«,

am,n,k,i = 0, l—k = m,n<l, all m and«,

am,n,k,i = 0,m<k, l^l^n, all wand«,

to any matrix of the type T we obtain a matrix of type 5 such that the

resulting transformation is identical with the original one. If for either

transformation ym,„ possesses a limit, the limit is called the generalized value

of the sequence xm,n by the transformation.

It is a well known fact that if a simple series converges, the corresponding

sequence in bounded. This need not hold for a double series. Thus consider

the series

«l,n = l,

«2.n=-l,

«m,n = 0, W^3.

This series converges, but the corresponding sequence is not bounded.

Thus convergent double series may be divided into two classes according

to whether the corresponding sequences are bounded or not. The following

definition for regularity of a transformation is constructed with regard to

a convergent bounded sequence ; thus even if a transformation is regular


1926] DIVERGENT DOUBLE SEQUENCES 53

it need not give to an unbounded convergent sequence the value to which

it converges.

If whenever xm,n is a bounded convergent sequence, ym,n converges to

the same value, then the transformation is said to be regular. A regular

transformation of real elements (am,B,A,i) is said to be totally regular,

provided when applied to a sequence of real elements (xm,n), which has the

following properties,

(a) xm,n is bounded for each m,

(b) xm,n is bounded for each »,

(c) lim xm,B = + oo,m,B—»oo

it transforms this sequence into a sequence which has for its limit + °°.

Concerning these transformations we shall prove necessary and sufficient

conditions for regularity, and then for total regularity. We shall give also

conditions which these definitions must satisfy in order that limm,B^ooym>B

shall exist, whenever limm,B„»a;m,n exists, irrespective of their values;

furthermore for the case where the sequence xm,n is merely bounded.

II. Regularity of linear transformations

The criterion for regularity of the transformations of type T is given

in the folloAving theorem:

Theorem I. A necessary and sufficient condition that any transformation

of type T be regular is

(a) lim am,n,k,i=0,for each k and I,ffl ,B-»00

m,B

(b) Hm E «m,B.A..= l,m,n->» A-l,f-l

m

(c) lim E \am,n,k.i\=0,for eachl,».«-»«o A—1

B

(d) lim E \am,n,k,i\ =0, for each k,m,n-»<» (_i

m,B

(«) E km.B.A.l|^4,t—1.1—1

where A is some constant.

Proof of necessity, (a) Define a sequence (xm,n) as follows: *m,B = l,

m = p, n = q; xm,n = 0, except when m = p, n = q. Then limm,B,<Da:m,B=0,

Vm.n amtn,P,q.



Hence in order that limm,„,a)ymin=0, it is necessary that limmi„.œam,„,p,s=0

for each p and q. Thus condition (a) is necessary.

(b) Consider the sequence (xm,n) defined as follows: xm,n = l. Then

ym,n= £"=i,;=i am,n.k,i. Since limm,n,„ ym,„ = l, condition (b) is necessary.

(c) To show the necessity of condition (c) we assume that condition (a)

is satisfied and that (c) is not, and obtain a contradiction. Since we are

assuming that for l = l0 (some fixed integer) the sequence Yl"=i\am.n.k,i\ does

not approach zero, for some preassigned constant h>0 there must exist a

sub-sequence of this sequence, such that each element of it is greater than h.

Choose Wi and «i such that

mi

2-1 l^mi.ni.Jt.Iol >A.t-1

Choose mi>m\\ n2>nx and such that

mi fo m3

2-1 lam2,n2,Mo I = ~, 2-1 lam2,n2,t,lol >A,*-i 2 k-i

and in general choose mp>mp-x ; np>np-i and such that

mp—1 fo mp

(1) 2-1 I amj>,»,,,*,loi < » 2_» lamr,np,*,!ol >A .*-l 2P *_,

From (1) we have

mp A / 1 \

(2) Z I«-»-,.m. I >*-— = *( 1-—-I.*— mp_i+l ¿ \ ¿ /

Define a sequence (a;m,n) as follows:

*m,n = 0, n¿¿l0 ;

xm,n = sgn am,,ni,k,i0, m=mi ;

xm,n = sgn am2i„2,*,io, mi<m-m2 ;

(3)

*m,n = Sgn amp,np,*,¡e, Wîp-i <»lá W,, ;

Here limmiI,.„ *m,n = 0. For this (xm,n) sequence we have

^mp.nj, — / . amp,np,k,lo Xk,lQ

*-l

»1,-1

— 2-t a">r>. "p.t.io **,lo~T 2j flmp,nB,*,Jo ^*,Io*—1 * = »„—,+!


1926] DIVERGENT DOUBLE SEQUENCES

From (1), (2), and (3) it follows that

55

• p-i

A-l

fnp-i

/ . ampinPlk,l<iZk.lo I = 7 . ^mp.np,fc,io

¡t-1 2p-i

k—mp_i+l fc—mp_i+l

anp,np.h.h =A( l~^lj-

Hence

^6*(1-¿)-¿-*('-¿)-

Thus ym,n does not have the limit zero, from which follows the necessity

of condition (e;).

(d) The above proof can be used for showing the necessity of condi-

tion (d) by simply interchanging the rôles of rows and columns.

(e) Assume conditions (a) and (b) are satisfied and that (e) does not

hold. Choose mx and nx such that

mi.fii

E lff»i,m,*i.l| =■ LA-l,(-l

Choose m-i>mx, «2>»i, and such that

mi.n,

E lamj,B2,*,.| =2 ,A—l.t-1

E \ami,m,k,l\ ^24 ,

and, in general, choose mv>mp-x, np>np-x and such that

(4)

m p—i,n p—i

E Um^n,,.*.. |=2"-\A—1,1-1

"E" I a«,,.,.».. 1*2**.A-l.t-l

From equations (4) we have

m p—i ,n p tn p,n p—\

E Iflm,B,*.ll+ E \amp,np,k,l |A-l,l-Bp_i+l *-m,_i+l, 1-1

+ "E" la^.n^.z | ^22"-2p-1^22p-22"-1 = 22"-1fc*-mp_i-f-l./=np_i+l



We now have two sequences of integers

Wíi< «tí < ms ■*£ wí< ' ■ * ,

»1< «2 < «J < «4 • • ' ,

such that

my—i,np_ i

£ \am„nT,t,i[^2*-\p>l,t-i,i-i

Wlp—l,flp Wl 4j|Wp—i

2-1 lflmp,n,,*,i| + 2-1 la<»j>,n,,*,l|*-l,!-np_jfl *-mp_i+l,i-l

(5)

+ "f *mp,np, ».il á?'-1t—mj-l+l.l—nj>_i+l

Define a sequence (a;m.„) as follows:

*m,n= Sgn ami,ni,t,i, ¿g>»,, /^«i,

*m,n= \ Sgn flm2,nj,*,i

(6)

m,n= r~I7Sgn am,,,np,*,l

nti<k^mi, l5=¿á»i

l^k = mi, mi<l^nt } ;

• tniKk^mz, »i</g«i

'mp-i<k^mP, lgi = »^i

lgA^«tp_i, »p_i</^«,

Here Kmm,„.„a;m,„ = 0. Consider

m p, n p mp—i ,np—i

ynp,np = ¿^ ömj,tnP,A,l^*,I= ¿^ amj>,nPli,J #*,!

Jt-1,/-I Jfe-l,I-l

Wy~l ifly fflp ,fl j>—1

+ 2^ <im,n,*,j ac*,i+ ¿J ûm))pBj>i)fc,ia;*,j*—l.i—n,_i+l Ic—mp^^l.l—l

+ 2-, Xlmp,np,k.lXk,l .k—mp_i+l,i-.np_i+l

From (5) and (6) we have

t-i,¡-i

i-,-iX . CLmv,nv,k,l Xk.l

m j-l,ny—i

= 2J \amr,np,k,li-l.i-1

S2»-1,


1926] DIVERGENT DOUBLE SEQUENCES

Mp—i,np in p,n p— i

fc—1,1—np_i+l *-"mp_i+l,i—1

+ "£'

57

*—mp-i+1, i—Bp_i+1

1 r "»p-i.»p,Bp,A,jXA.i=—— E lfl»p,

¿y La-1,!-B,_i+1

mp.n p—i mp,Bp

+ E |flmp,B,,A.l| + E |ûm,!t.,*,. |*— m,_ i+l.I— 1 A— mp_i+l,l— l J

f>p,k,l|

-2*r-l= 2»

Hence

Thus

I y«

2p-i

tp\^2'-2'-l = 2'-1

lim |ymp,Bp |=oo .

Since this sub-sequence of the (ym,B) sequence does not converge, the se-

quence (ym,n) has no limit and thus condition (e) is necessary.

Proof of sufficiency. Let the limit of the convergent sequence (xm,B)

be x; thenm.B

Vm,B— X = E am,n,*,!**,J — * ■*—l.t-1

From condition (¿>) we may write

(7)

where

Therefore

E am,n,*,J + ''iB,B= 1 ,t—l.t-i

lim rm,„ = 0.m,n-»o»

ym.n—X= E an,n,k,l(xk,l—x) — Tn,nX;

|ym.B-x| =

(8)

E am,n,k,l(Xk,l—x)k—l.t-1

+ E flm,B,*,l(**,t—*)*-!,(- q+1

+ \ È am,n,k.i(xk,i—x)I t-p+1,1-1

+ k«.B*|.

+ E am,B.*.t(**.i—*)»-P+l.t-ï+l



Since xm,»—*x, we can choose p and q so large that for any preassigned

small constant e

«|**.j—x\<-, wheneverk = p, l^q.

5A

Let L be the greatest of the numbers \xk,i—x\ for all k and /. Now choose

M and N such that whenever m = M, n — N, the following inequalities are

satisfied :

(Í) Z) l<*m.n,*.l|<—--k-i.i-i i>pqL

(n) <S\x\

(iii) 2Z |am.n,*.i|<— — ,1=1, 2, ■k-i 5qL

" e(iv) 2Z |ffm.n.*.i|<——-,Ä=1 , 2 ,

¡-i 5pL

Hence whenever m = M, n = N we have

e

\ym,n—x-X <-

SpqL-pqL +

5pL

e

,P

pL +

(from condition (a))

(from equation (7)),

(from condition (c)),

(from condition (d)) .

-qL5qL

Thus5¿ S\x\

lim ym,n-x=0 ,

X =i .

or ym,n—*x, which proves the theorem.

The following examples show that neither condition (c) nor condition

(d) follows from (e). We define a transformation of type T as follows:

1flm.n.t.i = —- •

2*M

Here

(a) lim am,n,k.i =0 ,»l,n-»ai

m.n n 1

(b) lim 2~2 am.n,k.i= lim 2~1- "1|m,n-<<o k—l,l—l m,n-»<» k—1 2k

1(c) lim Za<«,n,*,¡á lim —=0>

m ,n—*ro ¿—i m,n—» » W



1 1(d) lim E am,n,k.i = lim — — — **0 ,

m|B-»oo ibX m,n-*oo2 2

m,n m ,n

(«) E |am,B,*,i|= E <im,B,A,i = i.fc-1,1-1 *—1,1-1

If we consider the transformation (a»,,,j.i = l/2ffl), we shall find con-

ditions (a), (b), (c) and (e) satisfied, but not (d).

Let us further assume that the elements (am,„,k,i) of the transformation

are real and positive. We now have

E I am,n,*,í| = E am,n,k,l •*—1,1-1 *—l,t—1

It does not follow as in the case of the simple sequence that if linim,,,^

E*-u-i am,n,k,i = l, then E*=u=i am,n,k.i is bounded for all values of m

and n. Hence condition (e) of the preceding theorem does not follow from

condition (b). Thus we see that the criterion for regularity of a transforma-

tion is not simplified by making the above assumption.

Furthermore we have

m,n

Vm,B = E am,n,k,l Xk.l .*—1,1—1

Taking absolute values we obtain the following inequality:

m,n m.n

IVm.nlíi E | ffm,n.*,I**.t|á E \am,n,k.l \ K = A K ,A-1,1-1 A-l.i-1

where \xk,i\^K. Hence the

Corollary. Any bounded sequence (xm,n) is transformed by a regular

transformation of type T into a bounded sequence (ym,n).

Before considering the regularity of a transformation of type S, we will

prove certain lemmas.

Lemma I. //am,n^0, Em=î!n=i am,n diverges, it is possible to find a bounded

sequence em,n such that

(Í) 6m,B = 0,

(ii) lim em,B = 0 ,m,n—»»

CO

(iii) E ak,itk,i diverges.*-i,(-i

Proof. Let 5m,„= E*=",(=i <**.(• Therefore Sm+i.n+i&Sm.n+i>\n\Sm.n,

"Jm+l.n+l = "->m+l,n =Sm.n- ThUS linim,n-(» ■-> m.n = T" °° •



We define

{—- , if Sm.n^O ,

0, if5m,n=0.

Here

(i) «„,,» is bounded andèO ,

(Ü) lim im,n=0.m,n—.a>

From some value of m, « onward,

*m,nS= Ém+l,nS= *m+l,n+l , «m.n = im,n+l — em+li"+l •

For a fixed m, n, where €„,,„^0, we have

p,q m,q p,m

2-1 fl*.je*,«+ Z-i ak,Kk,i+ 2J «*.l«M*-m+l,l-n+l Jfc-l,I~n+l i-m+l,!-l

(9) r ™ m v 1Ê«m+p,n+8 Z^ a*,i "h 2-( Ot.l+ 2-r a*.»

Lt_m+1,/— n+1 *-l,i-n+l *-m+l,i-l -1

~ ^m+p,n+G\»Jm+p,n-r-g *^m,n/ •

In the above summation em+p,„+8 is in the smallest term except those which

are zero, but these terms drop out of the summation. Now choose p and q

so large that

1*Jm,n = ~~~ »^wi+p.n+fl ■

Then equation (9) becomes

p.g m,g p.n

2J dk.Kk.i + 2-, ak,itk,i+ 2J &k,itk,ii-m+l,J-n+l *-l.i-n+l ¡t-m+l,!-l

1 J_

Setting mx = p, «i=<7, this process can be repeated. Since this can be

carried out an infinite number of times, we have

CO t OB

Z flm.nim.n=00 •m—l,n—1

Lemma II. If Zw-û-i am,n is not absolutely convergent, it is possible

to choose a bounded sequence xm,n such that £,„,„—>0 and 2»-". n-i am,„xm,n

diverges to + <».



Proof. Under the hypothesis Em-M-i lam,B| diverges. Now choose

em,B as in the preceding lemma with regard to the series Em-r,B-i l«m,B|.

We define xm¡n = em,n sgn am,n; then am,nXm,n = am,nem,n sgn am,B = em,B|am,B|.

By the preceding lemma Em-In-i am,nXm,n diverges, as we wished to

prove.

We now proceed to consider transformations of type S.

Theorem II. In order that whenever a bounded sequence (x«,B) possesses

a limit x, E*°-i7i-i am,n,k.iXk.i shall converge and limm,B<œ E*°-û-i »m,.,t,i

• Xk,i—x, it is necessary and sufficient that

(a) lim am,n,k,i=0 for each k and I,m,it—too

00 ■ 00

(b) E I ûm,B,*,11 converge for each m and n,k-l,i-l

CO i CO

(c) lim E am,B,*,i=l ,*,*-»<»*-i,j_i

00

(d) Km E \am,n.k,i\=Ofor eachl,m,B-»«> A-l

00

(e) lim E Iam.n.t.tl =0/or each k ,»,«-»<• i-i

00 .00

(/*) E I «m.B,*,i I ÚA for all m and n .*—i.i—i

Proof of necessity, (a) Define a sequence (xm.n) as follows: x«.B = l,

m = p, « = ?; iCm,B = 0 except when m = p, n = q. Here ym,„ = am,B,A,l. Hence

condition (a) is necessary.

(b) Choose any fixed m and n and assume E*°=I,i-i |«m,B.*.t| diverges;

then there exists by the preceding Lemma II a bounded sequence xn,n

having the limit zero, and such that Er=ü-i am,n,k,iXk diverges. This

contradicts our hypothesis; hence condition (¿>) is necessary.

(c) Consider the sequence (xm,n) defined as follows: xm¡n = l, all m

and »; then ym,n = E*°-G-i am,n,k.r, and thus condition (e;) is necessary.

(d) To show the necessity of condition (d) we assume that conditions

(a) and (b) are satisfied but that (d) does not hold; and obtain a contra-

diction. Since the double sequence Er-i|0m,B,*,jo| (/o being a fixed integer)

does not approach zero, then for some preassigned small constant A>0

there must exist a sub-sequence of this sequence such that every element

of it is greater than h.



Choose mi, ni, and rx at random.

Choose 7»2>Wi, «2>«i such that

Ä h2J |a»>.»i.*.iol á— , from (a) ,

and r2>ri such that

2-1 lamj,ni,*,io| = h >jfc-l

00 /(

53 I «««.»«.Mol á— , from (6)k-rtf-l °

In general choose mp>mp-i, nP>np-i such that

(10)

S I ««„»„.Mo I á — , from (a) ,*-l o

00

Zj l«-»,>.np.Mol ^A !

and rp>rp_i such that

(ID ¿2 \amp,np.k.io |à — , from (ô)*-rp_i+l

From (10) and (11) we have

(12) ZJ l«m,,.»p.Mol è —h*-r,_l+l 4

Define a sequence (xm,n) as follows:

Xk.i = 0 , l^h ;

■ sgn aw,,M.*.io, Ä^ri ,

sgn cm2,„2,t,/0, ri<¿^r2 ,(13)

Xk.l-sgn a^,,,,^!,, »"p-^^á^p ,

Here limm,B<„ xk,i = 0. From (10) and (11),

'p-i

^ . amp,np,k,lo Xk.lq

r^ A

= 2-, l««p,»,,Mol = —t-l o

From (12) and (13),

' . amp,np,k,lo Xk.lo

-rp_l+l

= Z, l«»p.«,.*.iol =—ht-rp_i+l *



Consider

Iy«p.»pl ~ l 2^ ««„.»„.Mo^Mol*-i

= ¿3 l««p.«p,Mo*Mo| "■* Z3 ö«p,»p.Mo xMo*-r,-l+l *-l

7 • amp,npik,i(, xktit,k-rp+l

3 2h h= —h- — .

4 8 2

Thus limm,„,eo y™,» is not zero, hence (d) is necessary.

(e) In a similar manner we can show the necessity of (e).

(f) Assume conditions (a) and (b) are satisfied and that (f) does not

hold. Choose any mi, «i; r,, s, at random.

Choose w2>wi; «2>«i such that

13 |a««,»,,*,i|á2 , from (a) ;-l.i-i

7. |lm2,n2,t,l| =2* ■

k-l.l-1

Choose r2>fi; í2>íi such that

2-1 I «»«■»«,M I ~ 7 . | A««,»«,*-r2+l,¡=«2+l t—l.i-lH-1

+ E |a«2.n2,*.l|^22, from (b) .*_r2+l,i-l

Choose w3>w2; «3>«2 such that

* i

23 km8,n8,*,l| =22 ,-1.Í-1

00 I CO

23 |««8,»3,mI ^28 .*-l.!-l

Choose f3>r2, *3>$2 such that

CO .CD 00,58

7. |flm3,ns,t,l|+ Z-f | 0<n|,ns,*,l I*-rt+l,i-«s+l t-ra+l,i-l

+ 23 i««.,»..mI^2«.*—1,1-» !+l



In general, choose mp>mp-X, np>nP-x, such that

mp-i.Bp-i

E |amp.B,.*.i|=2*-1,4-l.t-l

(14)CP »00

E kmp.B„,*.t|=2,»;*—l.t-1

and rp>rp_i, sp>sp-X such that

OO ,00 m.'l

E |flm,,n„*.l| + E |flm„it„».l|*-Tp_1+l.l->p-l+l *-rp_i+l,(-l

(15)

+ E" Iflmp.^.tlá^-'.*-l,!-»p_l+l

From these inequalities we have

,"*' rr-i-sv **&->

E |flm„B„*.l| + E I flmp.it,.*.11 + E lflmp.it,.*.!!*-rp_i+l,l-«p_i+l r-l,!-jp_i+l *-r,_i+l.(-l

(16) ■£2i''—2%*-i-2*-l=2*-x[2»+l-2'~i-l]

^2"-1[2p+1—21"-1—2',-1] = 2,,,-l .

Define a sequence (xm,B) as follows:

x*,i = sgn flmi,Bt,*,i, k^mx , l£nx ;

*».!=—sgn am,.„,,A,i { mi<igm,, láí=?»i

(17)

lèk^mi, nx<l£n%

nx<k£mt, 1^/^*1

mx<k^mt, «i</^n»

f l^k^trtp-i, »p_i</^nr

f»pwi<*á»»p, K/á»»-i

Here lim,,,,,^, *m.„=0.

From the preceding inequalities (14), (15), (16) and (13) we hav?

/ lâ*amj-l| »p-K'â»,

**,!= -—^sgn am,,„,,A.! < mn-i<k^mp, Kl^np-i

(18)*,-l,«p-l mp-i.Bp-i

E am„B„*.i*A,tt á E |fl»,..,.».'lá2r"1 i*—l.l-l *-l,l-l



m p, n p m p—i ,fi p

23 «m,,»„*,l Xk,l + 23 fl«p.n,,M Xk,l*— my_i+l,l—np_i+l i—1,(—np_i+l

«p.»p—1 \ T~ «p,»p

(19) + 23 «»,.»,.m *m = —: 23 I««,,»p.mI*-«p_i+i.i-i ** L t-«p_i+i,i-»p_i+i

«p—l.»p mp.np -l

+ E I«»p.«,.mI+ 23 I««p.»p.mI £—2»>-i«2*;t-l,I-nP_,+ l t-«p_l+l,I-l J **^

k,l**.l + 23 ««p,np,*,l **,!Zrf fl«p,»p,t,I **.»T~ ¿^ >*«p,np,'*-mp+l,(-np+l *-l,l-nr+l

,»p

(20) + 23 ««p.»p.m*mk-mp+l.l-1

mp.nr,k,l\

«p,co

+ 23 |««p.»p.M*j-l,l-np+1 *- «p+1

1 r co .co

=-\ 23 I«2"L *-«,,+i,<-»p+i

+ Z' |««,,»,,M|lá¿2^-2-*

Consider

oa too

|y«p,»p| = | 23 ««p.»p.m*m|-1,1-1

mp,np mp-ipiip

A— mp_i+l, i—np_i+l *—1, i—np_i+l

Bip,n p—i

+ 23 ««p.Bp.M **.k—«p-1+1,1—1

23 fl«p.»pM *M*—«p+1, i—»p+1

m p, oo co ,np

+ 23 Om,.n,.k,lXk,l+ 23 a«,.»,.*.« **.«fc-l.I-np+1 *mp+l,l-l

«„¿»p

23 ««p.»p.M Xk.l»-1.1-1

è 2'-2"-1-2"-2=2',-*[4-2-l]

= 2*-2 , from (18) ,(19) ,(20) .

Hence limP.^Jym ,» | = + °°- Thus the sequence (y«,») has no finite limit,

hence (e) is necessary.

Proof of sufficiency. From definition we can write

03 tOO

y«.»—*= 23 ««,»,M *m—*•t—1,1—1

From condition (c) we have

23 ««.»,M+r«.«™l»t-i.i-i



where limm,B.œ f„,n = 0. Hence

co >eo

Vm.it-•*= E flm.Tt.*,! (Xk,l~ x)+rm,nX]*-l,!-l

P.S P.oo

|ym,n— X\ 2¡ | E flm.fi,*,! Xk,l | + | E flmp.n,,*,! **,! |*—1.1—1 *-l, 1-8+1

t» .8 00 .03

+ | E flm.n.t.l **,t I + I E flmp.n,,*,! **,! |+ l^m,«*!-A-p+1,1-1 *-jH-1,1-ï+l

Given e>0 we can choose p and q so large that

I**.i — *lá~—, when*>¿, l>q.5A

Let Z, be the greatest of the numbers |**,i—x\ for all k and /. Using con-

ditions (a), (d), and (e) we can choose two integers M and N such that

whenever m^ M,n^N,

(Í) E I flmp,Bp,*,I |<5/>gL

e(Ü) E 1 flmp,np,*.l|< - _ (*=1, 2, 3, • • • ,P) ,

i-i 0/>L

(iii) É |flm,.np,*,i|<—^— (/=1, 2, 3, • • • ,?);*=i oqL

(iv) |r«..|<51*|

We thus have, whenever m>M, n>N,

ly--xl=i^Lpq + jhLp + ~àa-L9

x\ + — A = t.S\x\' ' 5A

Hence

lim ym,n = x.m At-* a

Thus the theorem is proved.

From the equation

rm,n = E Am,»,*.! **,!



we have, taking absolute values,

oo i oo co t öd

IVm.nl =S E flm.it,*,! »*,I ̂ E I am,n.k.l \ K ̂ A K,*— l.t-1 A-l.I-1

where \xk,i\^K. Hence the

Corollary. A bounded sequence (xm,n) is transformed by a regular

transformation of type S into a bounded sequence (ym.n)-

If in the proof of the preceding theorem we replace the set of integers

1, 2, 3, • • • as range of variation of m by a point set T having a limiting

point t0 as range of variation of a variable t and the set of integers 1, 2, 3, ■ • •

as range of variations of » by a point set V having a limiting point v0 as range

of variation of a variable V, we shall have the proof of the folloAving general-

ization of the theorem.

Theorem III. Let ak,i(t, v) be defined for k = l, 2, 3, • • -, 1 = 1, 2, 3, • • • ,t in T, v in V, where T and V are two point sets in the real or complex plane,

having t0 and v0 (finite or infinite) respectively as limit points. Then in order

that whenever the bounded sequence (xm,n) possesses a finite limit x, it should

be true that for each pair of values t and v,

E fl*,i(', v) Xk.i exists1=1.1-1

and that

lim E a*.! (t, v) Xk,i = x ,<-»<•,»->»■ *—i,t-i

it is necessary and sufficient that

(a) lim Ok,i (t,v) = 0,foreachkandl;

oo .oo

(b) E I*1*.' ('» v) I converge for each t and v;t—l.t-i

oo .oo

(c) lim E •*.! (*»»)"!;t-.lt,«->»t A-1,1-1

00

(d) lim E I0*.' ('» ») I =0i for eacft h'-•'»,»-»»t *—i

00

(e) lim E I a*. i(t,v) |=0, for eachk;(->!t,f-'»t 1=1

00 .00

(/) E \ok.i(t, v)\< A for every pair of values of t and v.*-i.i-i



Theorem IV. Let ak,t(t, v) be defined for k ■■ 1, 2, 3, • • •, /■» 1, 2, 3, • • •,t in T, v in V, where T and V are two point sets in the real or complex plane,

having to and v¡> (finite or infinite) respectively as limit points. Then in order

that whenever the bounded sequence (xm>n) possesses a finite limit x, it shall

be true that for each pair of values t and v

OB lOD

23 ak,i(t, v)xk,i existst-i.i-i

and thatCO I co

lim 23 «m(*, ») Xk,i=\xi—t,,t—i, jt-i,i-i

(X being a fixed constant), it is necessary and sufficient that

lim Ok,i(t, v) = 0 for each k and l;(a) i-.»0,.-.t0

CO «OP

(b) 2w I«mC, ") I converge for each t and v;»-1,1-1

CO «op

(c) lim 23 <*k,i(t,v)=\;i-»/0>t-»t(, ».l.i-i

op

(d) lim 23 \ak,i(t,v)\=0 for each l;l-»«0,«-»»0 *-l

CO

(«) lim 23 I«m(*, p) I = 0, for each k;»"♦'Oi»-*»« l-l

co ,co

(J) 23 I«m(<» «0 I < A for all values oft and v.»-l.i-i

Proof. By forming a new transformation

. ,. v «m(*, «Obk.i(t, v)=-,

A

when X?*0, this problem reduces to that of the preceding theorem.

For X«=0 the proof is reduced to that of the preceding theorem by the

transformation defined as follows:

Let «i, «2, ««, ■ * • be a monotonically decreasing sequence of real num-

bers and such that eB—>0.

(a) If h and v<¡ are finite limit points, then for

€, ^ \t-tf, | + |t>- Do | < in-l, c - -f «>,

definebn,n(t, v)-a..n(t, v) + l,



ten, ten

bk.i(t,v) = ak,i(t,v) { k = n,ten

(b) If to is finite and v0 infinite, then for

1«.. á \t-k +T7<t»-i ' «0= + °°»

I» Idefine

bn.n(t, V)=an.n(t,v) + l,

(c) If to is infinite and v<> finite, then for

1«.. =î -T-r + |r-co|<iB-l, «o=+°°,

definebn.n(t, v) = a„,„(t, v)+l,

ten, l=n

ten, ten

bk,i(t,v) = ak,i(t,v) { k = n, ten }.

ten, ten

tem, ten

bk,i(t,v) = ak,i(t,v) { k=n,ten

ten, ten

(d) If h and v0 are both infinite, then for

1 1

M I» Idefine

bn,n(t, ») = flB.B(/, v) + l,

Íten, ten*=», ten

ten, ten

As defined, the transformation bk,i(t, v) is regular whenever the trans-

formation ak,i(t, v) satisfies the conditions of the theorem. If we write

00 • 00 00 t CO

y('>«0- E bk,l(t, v)Xk,l= E Ok,i(t, v)Xk,l + Xn.n,1=1,1-1 1=1,1=1

then by the preceding theorem

lim y(t, v) = lim E bk,i(t, «Ox*.i=»x.'-»'o.«-*»» <-»»o.»-*»» *—1.1-1



Thus

lim 23 «M (t, Xl)xk,i + X„,n \=X .

Hence00 ,00

lim 23 ak,i(t, v) xk,i+x = x,'->'o.»-,»o *—1,1=1

which reduces to the expression

00 ,0D

lim 23 ak,i(t, v) Xk,i = 0.t-.t0,v-*v0 »=1,1=1

Thus we have proved the theorem.

In the preceding work we assumed that the given sequence xm,n was

bounded and convergent and then found the necessary and sufficient con-

dition that the transformation must satisfy in order that the new sequence

ym,„ must converge to the same value. Different conditions may be placed

upon the given sequence xm,n and then we ask what conditions must the

transformation satisfy in order that the new sequence ym,n shall be bounded

and convergent. The results of these investigations are stated without

proof in the following theorems.

Theorem V. In order that a regular transformation of type T be totally

regular it is necessary and sufficient that there exist intergers ko and l0 such

that am,n,k,i>0 when k>k0, 1>U, for all values of m and n.

Theorem VI. A necessary and sufficient condition that a regular transforma-

tion of type T of real elements transform a bounded sequence (xm,„), whose

superior [inferior] limit is L [I], into a bounded sequence (ym.n) whose superior

[inferior] limit L' [I'] satisfies the relation L'^L [I'^l] is that the transforma-

tion be totally regular.

Theorem VIL A necessary and sufficient condition that a transformation

of type T transform a bounded convergent sequence (xm,n) into a bounded con-

vergent sequence (ym.n) is that the following conditions hold:

(a) lim a„,n,k,i exists for each k and I;«,»—»co

denote the value of the limit by C»,j;

m,n

(b) lim 23 ««,»,»,1 exists;mft-*<* t=i,l=l

denote the value of this limit by a;

«,n

(c) 23 \om.»,k.t\<A forollmondn;t-i.i-i



m

(d) lim E |flm,B,*,¡-Ci,¡ |=0, foreachl;m,lt->» A=i

n

(«) lim E |flm,it,*.i —c*.i|=0, for each k.m,n—K» i=X

When these conditions are satisfied, we have

00 iCO

lim ym,n = ax+ E c*.i(**,j—*),m,n-»» 1-1,1=1

îfAer« limm,„..„#*>,,,=£, <Ae series Er=u=i c*,i(~*,i—~0 ¿eiwg always con-

vergent.

Theorem VIII. A necessary and sufficient condition that the sequence

(Vm.n) defined by the relation

m,B

ym,n = E flm,n,l,l **,!1=1,1=1

shall converge whenever the bounded sequence (xm,n) converges and that the limit

of (ym.n) shall depend merely upon the limit of (xm¡„) and not upon the elements

of (xm,„), is that the following conditions hold:

(a) lim a-,«,i,i = 0, for each k and I;m,ti-»oo

m,B

(b) lim E flm,B,i,i exists;m,n-.oo a—1,1-1

m,B

(c) E |flm,n,i,i |<^4 for all m and n (A being a fixed con-1=1,1=1

stant) ;

m

(d) lim E |flm,B,i.i |=0, for eachl;m,n-><*> A_i

n

(e) lim E |flm,n,*,i 1 = 0, foreachk.m ,ii—»» j=i

II7 Ae« /Äese conditions are satisfied, we have

lim ym,B=aa;.m,■-♦«>

Theorem IX. A necessary and sufficient condition that a transformation

of the type S transform a bounded convergent sequence (xm,n) into a bounded

convergent sequence (ym.n) is that the following conditions hold:

(a) lim am,n,k,i exists for eachk and I;m,n—»«>



denote the value of the limit by ck,¡;

00 iQO

(b) lim 23 ««.»,*.i exists;»,»-»0» »=1,1=1

denote the value of this limit by a ;

CO «00

(c) 23 \am,„,k.i\<Aforallmandn;i=i,j=i

eo

(d) lira. 23 \am,n.k.i-Ck.i | = 0 for eachl;i>,»-» » *=i

0>

(e) lim 23 |«m.».*.*-c*,i|=0/oreacA*.«,»-.a> J=1


CO «CO

lim ym,n=ax+ 23 c*.i(**.i—*),*■»-»« *=i,i=i

where *=limm,„^0 *_,„, /Ae series 23r=u-i ¿».it**.«-*) fo»«g a/wayi ab-

solutely convergent.

Theorem X. A necessary and sufficient condition that the sequence (y-.,»)

defined by the relation

OO .00

y>n,n= ¿j dm.n.k.l Xk,l

»=1,1=1

shall converge whenever the bounded sequence (xm,n) converges, and that the

limit of (ym.n) shall depend merely upon the limit of (xm,„) and not upon the

elements of (xm,„), is that

(a) lim am,n,*,i=0 for each kandl,

00 ,00

(¿) 2v |««,»,t,i| converge for each m and n,t=i,i=i

CD «00

(c) Um 23 ««.».M exist,»■»-»" t=i,t=i

00

00 lim 23 |««,».*.i|=0, / any fixed integer,m,n—• a» £=i

OO

(«) lim 2~1 \am.fi.k.i\ = 0,k any fixed integer.


19261 DIVERGENT DOUBLE SEQUENCES 73


lim ym,n = ax .mjt-."

Theorem XL A necessary and sufficient condition that the transformation

(am,n,k,i) of type T transform a bounded sequence (xm,n) into a bounded con-

vergent sequence (ym,n) is that there exist numbers oa,i such that

m,tt

(0 lim E |flm,B,*,i — a*,i|=0,m--to» A=l,l=l

00 .00

(ii) E l-*.i| converges.*=i,i=i

Theorem XII. A necessary and sufficient condition that the transforma-

tion (am¡n,k,i) of type S transform a bounded sequence (xm,n) into a bounded

convergent sequence (ym,n) is that

oo too

(0 E |flm,it.*,!|*=1.I=1

converge for each m and « and that there exist numbers at.i such that

00 t CO

(Ü) Um E |flm,n,*.I —O*.l|=0,m ,!,-><■> A=l.l=l

00 .00

(iii) E a*.i converges.*=i.i=i

Cobneix University,Ithaca, N. Y.


divergent double sequences and series* · of the sequence xm,n by the transformation. it is a well...

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